Placing 4 Circles on a sphere
Facected regular solid &amp; boolean op.

Fastest way to find out how to sudbidive the 3D space of the sphere in n parts without relying on complicated solid-angle formulas is to use regular solids.

The idea behind this is to think about the sphere as only an "external shell" that defines a boundary; what we really care is to subdivide the space itself. Regular solids, because the way they are built, already have this property.

The operator can be found in the Extra Objects add-on, which is not enabled by default, but it comes boundled in with the official build.

As you would like to subdivide the sphere in 4 parts, what we are looking for is the Tetrahedron:



By simply scaling the solid, you can adjust the radius of the circles upon the sphere. At the moment there is no geometry on the common edge. In order to make the intersection real, we should also perform some kind of boolean operations.

In this example I added and apply an Intercept boolean modifier, than select the boundary edge loops with Select edge loop (Alt+MouseDx) and finished by running the Separate command.